I am proving, assuming this expression leads to something correct, that the tangent of a summation of angles is an expression involving sums of products of the tangents of each angle, like this: $$ \tan\left(\sum_{i=1}^{n}a_i\right)=\frac{\sum_{i_1=1}^n t_{i_1} - \sum_{i_1}^n ...\sum_{i_3}^{i_2} t_{i_1}\cdot...\cdot t_{i_3} + \sum_{i_1}^n\cdots\sum_{i_5}^{i_4} t_{i_1}\cdot...\cdot t_{i_5} -\cdots} { 1 - \sum_{i_1=1}^n \sum_{i_2=1}^{i_1} t_{i_1}\cdot t_{i_2}+ \sum_{i_1=1}^{n} ...\sum_{i_4=1}^{i_3} t_{i_1}\cdot...\cdot t_{i_4}-\cdots } $$
with $t_i=\tan(a_i)$, and the nested summation of products with a odd and even number of indices being at the numerator and denominator, respectively, alternating signs as the sum progress, and stopping the progression when there is no more indices left.
How could I improve this notation, using a better way to express the nested sums with products?